\babel@toc {english}{}\relax 
\contentsline {chapter}{Preface}{xv}{chapter*.4}%
\contentsline {chapter}{\numberline {1}Introduction and Foundations}{1}{chapter.1}%
\contentsline {section}{\numberline {1.1}Introduction}{1}{section.1.1}%
\contentsline {subsection}{\numberline {1.1.1}Motivation and Scope}{1}{subsection.1.1.1}%
\contentsline {subsection}{\numberline {1.1.2}Theoretical Challenges}{1}{subsection.1.1.2}%
\contentsline {section}{\numberline {1.2}System Architecture Overview}{2}{section.1.2}%
\contentsline {subsection}{\numberline {1.2.1}Core Components}{2}{subsection.1.2.1}%
\contentsline {subsection}{\numberline {1.2.2}Operational Characteristics}{3}{subsection.1.2.2}%
\contentsline {section}{\numberline {1.3}Mathematical Foundations}{3}{section.1.3}%
\contentsline {subsection}{\numberline {1.3.1}Notation and Conventions}{3}{subsection.1.3.1}%
\contentsline {subsection}{\numberline {1.3.2}Fundamental Concepts}{3}{subsection.1.3.2}%
\contentsline {subsection}{\numberline {1.3.3}System Dynamics}{4}{subsection.1.3.3}%
\contentsline {section}{\numberline {1.4}Problem Formulation}{4}{section.1.4}%
\contentsline {subsection}{\numberline {1.4.1}Objective Function}{4}{subsection.1.4.1}%
\contentsline {subsection}{\numberline {1.4.2}Constraints}{4}{subsection.1.4.2}%
\contentsline {section}{\numberline {1.5}Theoretical Contributions}{5}{section.1.5}%
\contentsline {subsection}{\numberline {1.5.1}Formal Models}{5}{subsection.1.5.1}%
\contentsline {subsection}{\numberline {1.5.2}Algorithmic Analysis}{5}{subsection.1.5.2}%
\contentsline {subsection}{\numberline {1.5.3}Performance Characterization}{5}{subsection.1.5.3}%
\contentsline {section}{\numberline {1.6}Chapter Organization}{5}{section.1.6}%
\contentsline {section}{\numberline {1.7}Prerequisites and Assumptions}{6}{section.1.7}%
\contentsline {section}{\numberline {1.8}Summary}{6}{section.1.8}%
\contentsline {chapter}{\numberline {2}Formal System Model}{7}{chapter.2}%
\contentsline {section}{\numberline {2.1}Introduction}{7}{section.2.1}%
\contentsline {section}{\numberline {2.2}Core POMDP Formulation}{7}{section.2.2}%
\contentsline {subsection}{\numberline {2.2.1}State Space Definition}{7}{subsection.2.2.1}%
\contentsline {subsection}{\numberline {2.2.2}Action Space Definition}{8}{subsection.2.2.2}%
\contentsline {subsection}{\numberline {2.2.3}Observation Model}{8}{subsection.2.2.3}%
\contentsline {subsection}{\numberline {2.2.4}Transition Dynamics}{9}{subsection.2.2.4}%
\contentsline {section}{\numberline {2.3}Hierarchical Options Framework}{9}{section.2.3}%
\contentsline {subsection}{\numberline {2.3.1}Options Definition}{9}{subsection.2.3.1}%
\contentsline {subsection}{\numberline {2.3.2}Option Selection Policy}{9}{subsection.2.3.2}%
\contentsline {section}{\numberline {2.4}Tool Environment Semantics}{10}{section.2.4}%
\contentsline {subsection}{\numberline {2.4.1}Tool Execution Model}{10}{subsection.2.4.1}%
\contentsline {subsection}{\numberline {2.4.2}Concurrent Tool Execution}{10}{subsection.2.4.2}%
\contentsline {section}{\numberline {2.5}Multi-Objective Optimization Framework}{10}{section.2.5}%
\contentsline {subsection}{\numberline {2.5.1}Objective Functions}{11}{subsection.2.5.1}%
\contentsline {subsection}{\numberline {2.5.2}Pareto Optimality}{11}{subsection.2.5.2}%
\contentsline {subsection}{\numberline {2.5.3}Scalarization and Solution Methods}{11}{subsection.2.5.3}%
\contentsline {section}{\numberline {2.6}Belief State Dynamics}{11}{section.2.6}%
\contentsline {subsection}{\numberline {2.6.1}Belief Update Mechanism}{11}{subsection.2.6.1}%
\contentsline {subsection}{\numberline {2.6.2}Information-Theoretic Analysis}{12}{subsection.2.6.2}%
\contentsline {section}{\numberline {2.7}Context Management Framework}{12}{section.2.7}%
\contentsline {subsection}{\numberline {2.7.1}Context State Representation}{12}{subsection.2.7.1}%
\contentsline {subsection}{\numberline {2.7.2}Context Selection Optimization}{12}{subsection.2.7.2}%
\contentsline {section}{\numberline {2.8}Learning and Adaptation}{13}{section.2.8}%
\contentsline {subsection}{\numberline {2.8.1}Online Learning Framework}{13}{subsection.2.8.1}%
\contentsline {subsection}{\numberline {2.8.2}Adaptive Parameter Tuning}{13}{subsection.2.8.2}%
\contentsline {section}{\numberline {2.9}System Properties and Guarantees}{13}{section.2.9}%
\contentsline {subsection}{\numberline {2.9.1}Stability and Convergence}{13}{subsection.2.9.1}%
\contentsline {subsection}{\numberline {2.9.2}Safety Guarantees}{14}{subsection.2.9.2}%
\contentsline {section}{\numberline {2.10}Computational Complexity}{14}{section.2.10}%
\contentsline {subsection}{\numberline {2.10.1}Decision Problem Complexity}{14}{subsection.2.10.1}%
\contentsline {subsection}{\numberline {2.10.2}Runtime Analysis}{14}{subsection.2.10.2}%
\contentsline {section}{\numberline {2.11}Summary}{14}{section.2.11}%
\contentsline {chapter}{\numberline {3}Mathematical Models and Algorithms}{17}{chapter.3}%
\contentsline {section}{\numberline {3.1}Introduction}{17}{section.3.1}%
\contentsline {section}{\numberline {3.2}Contextual Bandit Framework for Tool Selection}{17}{section.3.2}%
\contentsline {subsection}{\numberline {3.2.1}Problem Formulation}{17}{subsection.3.2.1}%
\contentsline {subsection}{\numberline {3.2.2}Linear Contextual Bandit Model}{18}{subsection.3.2.2}%
\contentsline {subsection}{\numberline {3.2.3}Thompson Sampling for Tool Selection}{18}{subsection.3.2.3}%
\contentsline {section}{\numberline {3.3}Submodular Optimization for Context Selection}{19}{section.3.3}%
\contentsline {subsection}{\numberline {3.3.1}Submodular Context Utility}{19}{subsection.3.3.1}%
\contentsline {subsection}{\numberline {3.3.2}Greedy Context Selection Algorithm}{19}{subsection.3.3.2}%
\contentsline {subsection}{\numberline {3.3.3}Dynamic Context Update}{19}{subsection.3.3.3}%
\contentsline {section}{\numberline {3.4}Queueing Models for Concurrency Management}{20}{section.3.4}%
\contentsline {subsection}{\numberline {3.4.1}System Model}{20}{subsection.3.4.1}%
\contentsline {subsection}{\numberline {3.4.2}Performance Analysis}{20}{subsection.3.4.2}%
\contentsline {subsection}{\numberline {3.4.3}Optimal Concurrency Level}{21}{subsection.3.4.3}%
\contentsline {section}{\numberline {3.5}Graph-Based Code Understanding}{21}{section.3.5}%
\contentsline {subsection}{\numberline {3.5.1}Code Dependency Graph}{21}{subsection.3.5.1}%
\contentsline {subsection}{\numberline {3.5.2}Exploration Strategy}{22}{subsection.3.5.2}%
\contentsline {section}{\numberline {3.6}Probabilistic Code Generation Model}{22}{section.3.6}%
\contentsline {subsection}{\numberline {3.6.1}Generation Framework}{22}{subsection.3.6.1}%
\contentsline {subsection}{\numberline {3.6.2}Constrained Decoding}{22}{subsection.3.6.2}%
\contentsline {subsection}{\numberline {3.6.3}Quality Estimation}{22}{subsection.3.6.3}%
\contentsline {section}{\numberline {3.7}Risk Assessment and Safety Models}{23}{section.3.7}%
\contentsline {subsection}{\numberline {3.7.1}Risk Quantification}{23}{subsection.3.7.1}%
\contentsline {subsection}{\numberline {3.7.2}Risk-Aware Decision Making}{23}{subsection.3.7.2}%
\contentsline {section}{\numberline {3.8}Online Learning and Adaptation}{24}{section.3.8}%
\contentsline {subsection}{\numberline {3.8.1}Multi-Armed Bandit with Side Information}{24}{subsection.3.8.1}%
\contentsline {subsection}{\numberline {3.8.2}Adaptive Parameter Tuning}{24}{subsection.3.8.2}%
\contentsline {section}{\numberline {3.9}Complexity Analysis}{24}{section.3.9}%
\contentsline {subsection}{\numberline {3.9.1}Time Complexity Results}{24}{subsection.3.9.1}%
\contentsline {subsection}{\numberline {3.9.2}Space Complexity Results}{25}{subsection.3.9.2}%
\contentsline {section}{\numberline {3.10}Summary}{25}{section.3.10}%
\contentsline {chapter}{\numberline {4}Algorithmic Analysis and Theoretical Guarantees}{27}{chapter.4}%
\contentsline {section}{\numberline {4.1}Introduction}{27}{section.4.1}%
\contentsline {section}{\numberline {4.2}Regret Analysis for Online Learning Components}{27}{section.4.2}%
\contentsline {subsection}{\numberline {4.2.1}Tool Selection Regret Bounds}{27}{subsection.4.2.1}%
\contentsline {subsection}{\numberline {4.2.2}Context Selection Regret}{28}{subsection.4.2.2}%
\contentsline {section}{\numberline {4.3}Convergence Analysis}{28}{section.4.3}%
\contentsline {subsection}{\numberline {4.3.1}Belief State Convergence}{28}{subsection.4.3.1}%
\contentsline {subsection}{\numberline {4.3.2}Multi-Objective Weight Adaptation}{28}{subsection.4.3.2}%
\contentsline {section}{\numberline {4.4}Approximation Algorithm Analysis}{29}{section.4.4}%
\contentsline {subsection}{\numberline {4.4.1}Context Selection Approximation Ratio}{29}{subsection.4.4.1}%
\contentsline {subsection}{\numberline {4.4.2}Code Exploration Approximation}{29}{subsection.4.4.2}%
\contentsline {section}{\numberline {4.5}Complexity Analysis}{29}{section.4.5}%
\contentsline {subsection}{\numberline {4.5.1}Computational Complexity Hierarchy}{29}{subsection.4.5.1}%
\contentsline {subsection}{\numberline {4.5.2}Runtime Analysis}{29}{subsection.4.5.2}%
\contentsline {section}{\numberline {4.6}Optimality Conditions}{30}{section.4.6}%
\contentsline {subsection}{\numberline {4.6.1}Pareto Optimality Characterization}{30}{subsection.4.6.1}%
\contentsline {subsection}{\numberline {4.6.2}Information-Theoretic Optimality}{30}{subsection.4.6.2}%
\contentsline {section}{\numberline {4.7}Stability Analysis}{31}{section.4.7}%
\contentsline {subsection}{\numberline {4.7.1}Queueing System Stability}{31}{subsection.4.7.1}%
\contentsline {subsection}{\numberline {4.7.2}Learning Algorithm Stability}{31}{subsection.4.7.2}%
\contentsline {section}{\numberline {4.8}Sample Complexity Analysis}{31}{section.4.8}%
\contentsline {subsection}{\numberline {4.8.1}PAC Learning Bounds}{31}{subsection.4.8.1}%
\contentsline {section}{\numberline {4.9}Robustness Analysis}{31}{section.4.9}%
\contentsline {subsection}{\numberline {4.9.1}Adversarial Robustness}{31}{subsection.4.9.1}%
\contentsline {subsection}{\numberline {4.9.2}Byzantine Fault Tolerance}{32}{subsection.4.9.2}%
\contentsline {section}{\numberline {4.10}Concentration Inequalities}{32}{section.4.10}%
\contentsline {subsection}{\numberline {4.10.1}Performance Concentration}{32}{subsection.4.10.1}%
\contentsline {section}{\numberline {4.11}Lower Bounds}{32}{section.4.11}%
\contentsline {subsection}{\numberline {4.11.1}Information-Theoretic Lower Bounds}{32}{subsection.4.11.1}%
\contentsline {subsection}{\numberline {4.11.2}Computational Lower Bounds}{32}{subsection.4.11.2}%
\contentsline {section}{\numberline {4.12}Adaptive Algorithm Analysis}{33}{section.4.12}%
\contentsline {subsection}{\numberline {4.12.1}Parameter Adaptation Rates}{33}{subsection.4.12.1}%
\contentsline {section}{\numberline {4.13}Statistical Efficiency}{33}{section.4.13}%
\contentsline {subsection}{\numberline {4.13.1}Minimax Optimality}{33}{subsection.4.13.1}%
\contentsline {section}{\numberline {4.14}Summary}{33}{section.4.14}%
\contentsline {chapter}{\numberline {5}Performance Analysis and Optimization}{35}{chapter.5}%
\contentsline {section}{\numberline {5.1}Introduction}{35}{section.5.1}%
\contentsline {section}{\numberline {5.2}Multi-Objective Performance Metrics}{35}{section.5.2}%
\contentsline {subsection}{\numberline {5.2.1}Performance Metric Definitions}{35}{subsection.5.2.1}%
\contentsline {subsection}{\numberline {5.2.2}Pareto Efficiency Analysis}{35}{subsection.5.2.2}%
\contentsline {section}{\numberline {5.3}Scalability Analysis}{36}{section.5.3}%
\contentsline {subsection}{\numberline {5.3.1}Computational Scaling}{36}{subsection.5.3.1}%
\contentsline {subsection}{\numberline {5.3.2}Memory Scaling}{36}{subsection.5.3.2}%
\contentsline {section}{\numberline {5.4}Adaptive Optimization Strategies}{36}{section.5.4}%
\contentsline {subsection}{\numberline {5.4.1}Dynamic Parameter Tuning}{36}{subsection.5.4.1}%
\contentsline {section}{\numberline {5.5}Load Balancing and Resource Management}{36}{section.5.5}%
\contentsline {section}{\numberline {5.6}Summary}{36}{section.5.6}%
\contentsline {chapter}{\numberline {6}Safety, Security and Verification}{37}{chapter.6}%
\contentsline {section}{\numberline {6.1}Introduction}{37}{section.6.1}%
\contentsline {section}{\numberline {6.2}Safety Property Specification}{37}{section.6.2}%
\contentsline {subsection}{\numberline {6.2.1}Safety Invariants}{37}{subsection.6.2.1}%
\contentsline {subsection}{\numberline {6.2.2}Temporal Logic Specifications}{37}{subsection.6.2.2}%
\contentsline {section}{\numberline {6.3}Formal Verification Methods}{37}{section.6.3}%
\contentsline {subsection}{\numberline {6.3.1}Model Checking Approach}{37}{subsection.6.3.1}%
\contentsline {subsection}{\numberline {6.3.2}Theorem Proving for Complex Properties}{38}{subsection.6.3.2}%
\contentsline {section}{\numberline {6.4}Security Analysis Framework}{38}{section.6.4}%
\contentsline {subsection}{\numberline {6.4.1}Threat Model}{38}{subsection.6.4.1}%
\contentsline {subsection}{\numberline {6.4.2}Security Guarantees}{38}{subsection.6.4.2}%
\contentsline {section}{\numberline {6.5}Risk Assessment and Management}{38}{section.6.5}%
\contentsline {subsection}{\numberline {6.5.1}Probabilistic Risk Models}{38}{subsection.6.5.1}%
\contentsline {subsection}{\numberline {6.5.2}Defense in Depth}{38}{subsection.6.5.2}%
\contentsline {section}{\numberline {6.6}Verification Tools and Methods}{38}{section.6.6}%
\contentsline {subsection}{\numberline {6.6.1}Static Analysis}{38}{subsection.6.6.1}%
\contentsline {subsection}{\numberline {6.6.2}Runtime Monitoring}{38}{subsection.6.6.2}%
\contentsline {section}{\numberline {6.7}Summary}{39}{section.6.7}%
\contentsline {chapter}{\numberline {7}Empirical Validation and Applications}{41}{chapter.7}%
\contentsline {section}{\numberline {7.1}Introduction}{41}{section.7.1}%
\contentsline {section}{\numberline {7.2}Benchmarking Framework}{41}{section.7.2}%
\contentsline {subsection}{\numberline {7.2.1}Benchmark Suite Design}{41}{subsection.7.2.1}%
\contentsline {subsection}{\numberline {7.2.2}Evaluation Metrics}{41}{subsection.7.2.2}%
\contentsline {section}{\numberline {7.3}Statistical Testing Framework}{41}{section.7.3}%
\contentsline {subsection}{\numberline {7.3.1}Hypothesis Testing}{41}{subsection.7.3.1}%
\contentsline {subsection}{\numberline {7.3.2}Confidence Intervals}{42}{subsection.7.3.2}%
\contentsline {section}{\numberline {7.4}Experimental Design}{42}{section.7.4}%
\contentsline {subsection}{\numberline {7.4.1}A/B Testing Framework}{42}{subsection.7.4.1}%
\contentsline {subsection}{\numberline {7.4.2}Sample Size Determination}{42}{subsection.7.4.2}%
\contentsline {section}{\numberline {7.5}Case Studies}{42}{section.7.5}%
\contentsline {subsection}{\numberline {7.5.1}Real-World Deployment Analysis}{42}{subsection.7.5.1}%
\contentsline {subsection}{\numberline {7.5.2}Performance Validation}{42}{subsection.7.5.2}%
\contentsline {section}{\numberline {7.6}Deployment Guidelines}{43}{section.7.6}%
\contentsline {subsection}{\numberline {7.6.1}Best Practices}{43}{subsection.7.6.1}%
\contentsline {subsection}{\numberline {7.6.2}Configuration Recommendations}{43}{subsection.7.6.2}%
\contentsline {section}{\numberline {7.7}Future Research Directions}{43}{section.7.7}%
\contentsline {subsection}{\numberline {7.7.1}Theoretical Extensions}{43}{subsection.7.7.1}%
\contentsline {subsection}{\numberline {7.7.2}Practical Applications}{43}{subsection.7.7.2}%
\contentsline {section}{\numberline {7.8}Summary}{43}{section.7.8}%
\contentsline {chapter}{Appendix Mathematical Proofs}{45}{appendix*.16}%
\contentsline {section}{\numberline {A.1}Chapter 2 Proofs}{45}{section.1..1}%
\contentsline {subsection}{\numberline {A.1.1}Proof of Theorem \ref {thm:hierarchical_improvement}}{45}{subsection.1..1.1}%
\contentsline {subsection}{\numberline {A.1.2}Proof of Theorem \ref {thm:pareto_set}}{45}{subsection.1..1.2}%
\contentsline {section}{\numberline {A.2}Chapter 3 Proofs}{46}{section.1..2}%
\contentsline {subsection}{\numberline {A.2.1}Proof of Theorem \ref {thm:linucb_regret}}{46}{subsection.1..2.1}%
\contentsline {section}{\numberline {A.3}Chapter 4 Proofs}{46}{section.1..3}%
\contentsline {subsection}{\numberline {A.3.1}Proof of Theorem \ref {thm:greedy_optimal}}{46}{subsection.1..3.1}%
\contentsline {section}{\numberline {A.4}Information-Theoretic Proofs}{47}{section.1..4}%
\contentsline {subsection}{\numberline {A.4.1}Mutual Information Properties}{47}{subsection.1..4.1}%
\contentsline {section}{\numberline {A.5}Queueing Theory Proofs}{47}{section.1..5}%
\contentsline {subsection}{\numberline {A.5.1}Proof of Stability Conditions}{47}{subsection.1..5.1}%
\contentsline {section}{\numberline {A.6}Concentration Inequality Proofs}{48}{section.1..6}%
\contentsline {subsection}{\numberline {A.6.1}Proof of Performance Concentration}{48}{subsection.1..6.1}%
\contentsline {section}{\numberline {A.7}Robustness Proofs}{48}{section.1..7}%
\contentsline {subsection}{\numberline {A.7.1}Proof of Distribution Robustness}{48}{subsection.1..7.1}%
\contentsline {section}{\numberline {A.8}Lower Bound Proofs}{48}{section.1..8}%
\contentsline {subsection}{\numberline {A.8.1}Proof of Information-Theoretic Lower Bounds}{48}{subsection.1..8.1}%
\contentsline {section}{\numberline {A.9}Complexity Theory Proofs}{48}{section.1..9}%
\contentsline {subsection}{\numberline {A.9.1}Proof of PSPACE-Hardness}{48}{subsection.1..9.1}%
\contentsline {section}{\numberline {A.10}Statistical Learning Theory Proofs}{49}{section.1..10}%
\contentsline {subsection}{\numberline {A.10.1}PAC Learning Bounds}{49}{subsection.1..10.1}%
\contentsline {chapter}{Appendix Algorithm Specifications}{51}{appendix*.17}%
\contentsline {section}{\numberline {B.11}Online Learning Algorithms}{51}{section.1..11}%
\contentsline {subsection}{\numberline {B.11.1}LinUCB for Tool Selection}{51}{subsection.1..11.1}%
\contentsline {subsection}{\numberline {B.11.2}Thompson Sampling with Gaussian Priors}{52}{subsection.1..11.2}%
\contentsline {section}{\numberline {B.12}Context Selection Algorithms}{52}{section.1..12}%
\contentsline {subsection}{\numberline {B.12.1}Greedy Submodular Maximization}{52}{subsection.1..12.1}%
\contentsline {subsection}{\numberline {B.12.2}Dynamic Context Update}{53}{subsection.1..12.2}%
\contentsline {section}{\numberline {B.13}Code Exploration Algorithms}{54}{section.1..13}%
\contentsline {subsection}{\numberline {B.13.1}Priority-Based Graph Exploration}{54}{subsection.1..13.1}%
\contentsline {section}{\numberline {B.14}Risk Assessment Algorithms}{55}{section.1..14}%
\contentsline {subsection}{\numberline {B.14.1}Risk-Aware Action Selection}{55}{subsection.1..14.1}%
\contentsline {section}{\numberline {B.15}Concurrency Control Algorithms}{56}{section.1..15}%
\contentsline {subsection}{\numberline {B.15.1}Adaptive Concurrency Management}{56}{subsection.1..15.1}%
\contentsline {section}{\numberline {B.16}Multi-Objective Optimization Algorithms}{57}{section.1..16}%
\contentsline {subsection}{\numberline {B.16.1}Pareto Frontier Approximation}{57}{subsection.1..16.1}%
\contentsline {section}{\numberline {B.17}Belief State Management}{58}{section.1..17}%
\contentsline {subsection}{\numberline {B.17.1}Particle Filter for Belief Updates}{58}{subsection.1..17.1}%
\contentsline {section}{\numberline {B.18}Implementation Notes}{58}{section.1..18}%
\contentsline {subsection}{\numberline {B.18.1}Computational Optimizations}{58}{subsection.1..18.1}%
\contentsline {subsection}{\numberline {B.18.2}Numerical Stability}{58}{subsection.1..18.2}%
\contentsline {chapter}{Appendix Mathematical Foundations}{59}{appendix*.18}%
\contentsline {section}{\numberline {C.19}Probability Theory Basics}{59}{section.1..19}%
\contentsline {subsection}{\numberline {C.19.1}Probability Spaces and Measure Theory}{59}{subsection.1..19.1}%
\contentsline {subsection}{\numberline {C.19.2}Random Variables and Distributions}{59}{subsection.1..19.2}%
\contentsline {subsection}{\numberline {C.19.3}Concentration Inequalities}{60}{subsection.1..19.3}%
\contentsline {section}{\numberline {C.20}Linear Algebra}{60}{section.1..20}%
\contentsline {subsection}{\numberline {C.20.1}Vector Spaces and Linear Maps}{60}{subsection.1..20.1}%
\contentsline {subsection}{\numberline {C.20.2}Matrix Theory}{61}{subsection.1..20.2}%
\contentsline {subsection}{\numberline {C.20.3}Matrix Calculus}{61}{subsection.1..20.3}%
\contentsline {section}{\numberline {C.21}Convex Analysis}{61}{section.1..21}%
\contentsline {subsection}{\numberline {C.21.1}Convex Sets}{61}{subsection.1..21.1}%
\contentsline {subsection}{\numberline {C.21.2}Convex Functions}{62}{subsection.1..21.2}%
\contentsline {section}{\numberline {C.22}Information Theory Basics}{62}{section.1..22}%
\contentsline {subsection}{\numberline {C.22.1}Entropy and Mutual Information}{62}{subsection.1..22.1}%
\contentsline {subsection}{\numberline {C.22.2}Divergences and Distances}{63}{subsection.1..22.2}%
\contentsline {section}{\numberline {C.23}Graph Theory Fundamentals}{63}{section.1..23}%
\contentsline {subsection}{\numberline {C.23.1}Basic Definitions}{63}{subsection.1..23.1}%
\contentsline {subsection}{\numberline {C.23.2}Graph Algorithms}{63}{subsection.1..23.2}%
\contentsline {subsection}{\numberline {C.23.3}Centrality Measures}{64}{subsection.1..23.3}%
\contentsline {section}{\numberline {C.24}Asymptotic Notation}{64}{section.1..24}%
\contentsline {chapter}{Appendix Statistical Learning Theory}{65}{appendix*.19}%
\contentsline {section}{\numberline {D.25}PAC Learning Framework}{65}{section.1..25}%
\contentsline {subsection}{\numberline {D.25.1}Basic Setup}{65}{subsection.1..25.1}%
\contentsline {subsection}{\numberline {D.25.2}Fundamental Results}{65}{subsection.1..25.2}%
\contentsline {section}{\numberline {D.26}VC Dimension and Sample Complexity}{66}{section.1..26}%
\contentsline {subsection}{\numberline {D.26.1}VC Dimension}{66}{subsection.1..26.1}%
\contentsline {subsection}{\numberline {D.26.2}Generalization Bounds}{66}{subsection.1..26.2}%
\contentsline {section}{\numberline {D.27}Concentration Inequalities}{67}{section.1..27}%
\contentsline {subsection}{\numberline {D.27.1}Basic Concentration Results}{67}{subsection.1..27.1}%
\contentsline {subsection}{\numberline {D.27.2}Martingale-Based Inequalities}{67}{subsection.1..27.2}%
\contentsline {section}{\numberline {D.28}Online Learning and Regret Bounds}{67}{section.1..28}%
\contentsline {subsection}{\numberline {D.28.1}Online Learning Framework}{67}{subsection.1..28.1}%
\contentsline {subsection}{\numberline {D.28.2}Gradient Descent in Online Setting}{68}{subsection.1..28.2}%
\contentsline {subsection}{\numberline {D.28.3}Follow-the-Regularized-Leader}{68}{subsection.1..28.3}%
\contentsline {section}{\numberline {D.29}Multi-Armed Bandits Fundamentals}{68}{section.1..29}%
\contentsline {subsection}{\numberline {D.29.1}Stochastic Multi-Armed Bandits}{68}{subsection.1..29.1}%
\contentsline {subsection}{\numberline {D.29.2}Upper Confidence Bounds}{68}{subsection.1..29.2}%
\contentsline {subsection}{\numberline {D.29.3}Lower Bounds}{69}{subsection.1..29.3}%
\contentsline {subsection}{\numberline {D.29.4}Contextual Bandits}{69}{subsection.1..29.4}%
\contentsline {section}{\numberline {D.30}Learning in Code Intelligence Systems}{69}{section.1..30}%
\contentsline {subsection}{\numberline {D.30.1}Tool Selection as Bandit Problem}{69}{subsection.1..30.1}%
\contentsline {subsection}{\numberline {D.30.2}Sample Complexity for Code Understanding}{70}{subsection.1..30.2}%
\contentsline {subsection}{\numberline {D.30.3}Online Learning for Dynamic Environments}{70}{subsection.1..30.3}%
\contentsline {chapter}{Appendix Optimization Theory}{71}{appendix*.20}%
\contentsline {section}{\numberline {E.31}Convex Optimization Basics}{71}{section.1..31}%
\contentsline {subsection}{\numberline {E.31.1}Convex Optimization Problems}{71}{subsection.1..31.1}%
\contentsline {subsection}{\numberline {E.31.2}First-Order Conditions}{71}{subsection.1..31.2}%
\contentsline {section}{\numberline {E.32}KKT Conditions and Duality}{72}{section.1..32}%
\contentsline {subsection}{\numberline {E.32.1}Karush-Kuhn-Tucker Conditions}{72}{subsection.1..32.1}%
\contentsline {subsection}{\numberline {E.32.2}Lagrangian Duality}{72}{subsection.1..32.2}%
\contentsline {section}{\numberline {E.33}Gradient Descent Methods}{72}{section.1..33}%
\contentsline {subsection}{\numberline {E.33.1}Basic Gradient Descent}{72}{subsection.1..33.1}%
\contentsline {subsection}{\numberline {E.33.2}Accelerated Gradient Methods}{73}{subsection.1..33.2}%
\contentsline {subsection}{\numberline {E.33.3}Stochastic Gradient Descent}{73}{subsection.1..33.3}%
\contentsline {section}{\numberline {E.34}Submodular Function Theory}{73}{section.1..34}%
\contentsline {subsection}{\numberline {E.34.1}Submodular Functions}{73}{subsection.1..34.1}%
\contentsline {subsection}{\numberline {E.34.2}Submodular Maximization}{74}{subsection.1..34.2}%
\contentsline {subsection}{\numberline {E.34.3}Non-Monotone Submodular Maximization}{74}{subsection.1..34.3}%
\contentsline {section}{\numberline {E.35}Multi-Objective Optimization}{74}{section.1..35}%
\contentsline {subsection}{\numberline {E.35.1}Pareto Optimality}{74}{subsection.1..35.1}%
\contentsline {subsection}{\numberline {E.35.2}Scalarization Methods}{75}{subsection.1..35.2}%
\contentsline {section}{\numberline {E.36}Applications to Code Intelligence}{75}{section.1..36}%
\contentsline {subsection}{\numberline {E.36.1}Tool Selection as Multi-Objective Optimization}{75}{subsection.1..36.1}%
\contentsline {subsection}{\numberline {E.36.2}Context Selection as Submodular Optimization}{75}{subsection.1..36.2}%
\contentsline {subsection}{\numberline {E.36.3}Online Optimization for Dynamic Tool Selection}{75}{subsection.1..36.3}%
\contentsline {section}{\numberline {E.37}Advanced Topics}{76}{section.1..37}%
\contentsline {subsection}{\numberline {E.37.1}Non-Convex Optimization}{76}{subsection.1..37.1}%
\contentsline {subsection}{\numberline {E.37.2}Distributionally Robust Optimization}{76}{subsection.1..37.2}%
\contentsline {chapter}{Bibliography}{77}{appendix*.21}%
